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Higher Derivative Corrections to the Abelian Born-Infeld Action using Superspace Methods

vrijdag, 22 september, 2006 - 15:30
Campus: Brussels Humanities, Sciences & Engineering campus
Faculteit: Science and Bio-engineering Sciences
Stijn Nevens

All the efforts done in theoretical physics today can grosso modo be split into
two categories. The first is finding the fundamental underlying principles that
govern physical phenomena and the second is trying to explain what we observe
in nature starting from these principles. The quest for a theory of everything is
primordially an effort find a single underlying principle. At the moment there
are two complementary theories which combined give an excellent description of
nature: the Standard Model and General Relativity. From this it is apparent
that more work is needed towards a single underlying theory. In the ideal case
we need a theory that unifies gravity with the other forces and correctly predicts
the twenty-six free parameters in the Standard Model. A promising candidate is
String Theory in which one assumes that elementary particles are small vibra-
ting strings rather than point-particles. In the low energy regime, the resulting
model is essentially a supersymmetric version of general relativity coupled to a
supersymmetric gauge theory.

Although string theory has many pleasing features it also is spawned with
various connected problems. Efforts to solve these naturally opened up a the door
to the discovery of D-branes which revolutionized string theory. A Dp-brane is a
p-dimensional dynamical object defined by the fact that open strings can end on
it. A tantalizing aspect of D-branes is their intimate relation with gauge theories.
Indeed, the effective action of a single brane in the slowly varying field limit is
the Born-Infeld action which in leading order is nothing more than the action
for ordinary Maxwell theory. This is the result not including derivatives on the
fieldstrength of the U(1) gauge field. It was shown that the term containing
two derivatives vanishes and it was not until recently that the four derivative
corrections where calculated.

In this thesis we study two-dimensional supersymmetric non-linear sigma-
models with boundaries. We derive the most general family of boundary condi-
tions in the non-supersymmetric case. Next we show that no further conditions
arise when passing to the N = 1 model and we present a manifest N = 1 off-shell
formulation. The analysis is greatly simplified compared to previous studies and
there is no need to introduce non-local superspaces nor to go (partially) on-shell.
Whether or not torsion is present does not modify the discussion. Subsequently,
we determine under which conditions a second supersymmetry exists. As for the
case without boundaries, two covariantly constant complex structures are needed.
However, because of the presence of the boundary, one gets expressed in terms of the other one and the remainder of the geometric data. Finally we recast some
of our results in N = 2 superspace.

Leaning on these results we then calculate the beta-functions through three
loops for an open string sigma-model in the presence of U(1) background. Re-
quiring them to vanish is then reinterpreted as the string equations of motion
for the background. Upon integration this yields the low energy effective action.
Doing the calculation in N = 2 boundary superspace significantly simplifies the
calculation. The one loop contribution gives the effective action to all orders
in alpha’ in the limit of a constant fieldstrength. The result is the well known
Born-Infeld action. The absence of a two loop contribution to the beta-function
shows the absence of two derivative terms in the action. Finally the three loop
contribution gives the four derivative terms in the effective action to all orders in
alpha’. Modulo a field redefinition we find complete agreement with the proposal
made in the literature. By doing the calculation in N = 2 superspace, we get a
nice geometric characterization of UV finiteness of the non-linear sigma-model:
UV finiteness is guaranteed provided that the background is a deformed stable
holomorphic bundle.